Characterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaces

Abstract

Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L} = -\Delta + V$ acting on $L^2(\mathbb{R}^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q \geq n$. Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0 \leq \lambda \lt n$ denote the Morrey space on $\mathbb{R}^{n}$. In this paper, we will show that a function $f \in L^{2,\lambda}(\mathbb{R}^{n})$ is the trace of the solution of $\mathbb{L}u := u_{t} + \mathcal{L}u = 0$, $u(x,0) = f(x)$, where $u$ satisfies a Carleson-type condition \[ \sup_{x_B,r_B} r_B^{-\lambda} \int_0^{r_B^2} \!\! \int_{B(x_B,r_B)} |\nabla u(x,t)|^2 \, dx dt \leq C \lt \infty. \] Conversely, this Carleson-type condition characterizes all the $\mathbb{L}$-carolic functions whose traces belong to the Morrey space $L^{2,\lambda}(\mathbb{R}^{n})$ for all $0 \leq \lambda \lt n$. This result extends the analogous characterization found by Fabes and Neri in [8] for the classical BMO space of John and Nirenberg.